Question

A body cools from a temperature $3T$ to $2T$ in 10 minutes. The room temperature is $T$. Assume that Newton’s law of cooling is applicable. The temperature of the body at the end of next 10 minutes will be:
A. $\dfrac{7}{4}T$
B. $\dfrac{3}{2}T$
C. $\dfrac{4}{3}T$
D. $T$

Answer 1

Hint: To find the temperature of the body which is cooling with time we use Newton’s law of cooling. We have to assume that the temperature of the surroundings is constant.

Complete step by step solution:
According to Newton’s law of cooling the rate of change of temperature of a body through radiation is directly proportional to the difference in the temperature of the body and the surrounding.
In mathematical form, the Newton’s law of cooling is given as,
$\begin{align}
  & \dfrac{dT}{dt}=k\left( {{T}_{t}}-{{T}_{s}} \right) \\
 & \text{where,} \\
 & k=\text{Newton }\!\!'\!\!\text{ s cooling constant} \\
 & {{T}_{t}}=\text{ temperature of the body at any time }t \\
 & {{T}_{s}}=\text{ temperature of the surrounding } \\
\end{align}$
On simplifying the Newton’s law of cooling, we get,
$\ln \left( \dfrac{{{T}_{2}}-{{T}_{s}}}{{{T}_{1}}-T} \right)=kt$
For the first 10mins
$\begin{align}
  & \ln \left( \dfrac{3T-{{T}_{s}}}{2T-T} \right)=k\left( 10 \right) \\
 & \ln \left( \dfrac{2T}{T} \right)=10k \\
 & \ln 2=10k\ldots \ldots \left( i \right)
\end{align}$

For next 10 mins,
Let the temperature at the end of next 10 mins is $T'$
Then,
$\begin{align}
  & \ln \left( \dfrac{2T-{{T}_{s}}}{T'-T} \right)=k\left( 10 \right) \\
 & \ln \left( \dfrac{T}{T'-T} \right)=10k\ldots \ldots \left( ii \right)
\end{align}$

Dividing equation $\left( i \right)$by equation$\left( ii \right)$we get,
$\begin{align}
  & \dfrac{\ln 2}{\ln \left( \dfrac{T}{T'-T} \right)}=\dfrac{10k}{10k} \\
& \Rightarrow \ln \left( \dfrac{T}{T'-T} \right)=\ln 2 \\
& \Rightarrow \dfrac{T}{T'-T}=2 \\
& \Rightarrow 2T'-2T=T \\
& \Rightarrow 2T'=3T \\
& \Rightarrow T'=\dfrac{3}{2}T
\end{align}$

Hence, the temperature of the body after next 10 mins is $\dfrac{3}{2}T$.

Note: We assume the temperature of the room is constant. The body cools down by radiating out the heat energy contained in it.